Differentiate the following function with respect to x: (log x)x+x(logx) - Mathematics

Question

Differentiate the following function with respect to x: ${(\log x)}^{x} + x^{\log x}$

Solution

$L e t y = {(\log x)}^{x} + x^{\log x} ... ... ... ... . (1)$

$N o w$

$y = y_{1} + y_{2} ... ... ... ... ... ... ... ... . . (2)$

Differentiating (2) with respect x, we get

$\frac{d y}{d x} = \frac{{d y}_{1}}{d x} + \frac{{d y}_{2}}{d x} ... ... ... (3)$

Now take log of y₁ = (log x)^x

${\log y}_{1} = x \log (\log x)$

Differentiating with respect to x, we get

$\frac{1}{y_{2}} \frac{{d y}_{2}}{d x} = (2 \log x) \times \frac{1}{x}$

$\frac{{d y}_{2}}{d x} = y_{2} (\frac{2 \log x}{x}) = x^{\log x} (\frac{2 \log x}{x}) ... ... ... ... ... . (5)$

Adding equation (4) and (5), we get:

$\frac{d y}{d x} = {(\log x)}^{x} (\frac{1}{\log x} + \log (\log x)) + x^{\log x} (\frac{2 \log x}{x})$

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Logarithmic Differentiation

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Q 14 Q 13 Q 15

2012-2013 (March) Delhi Set 1

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Differentiate the following function with respect to x: (log x)x+x(logx) - Mathematics

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